As stellar systems, such polytropes arise as spheres with isotropic velocity dispersions, with a phase-space distribution function, \(f\) , that depends on velocity and position only through the energy, \(f({\bf r}, {\bf v})=f(E)\) , where \(E={\bf v}^2/2+\phi \)  (\(\phi \)  is the gravitational potential), and, furthermore \(f(E)\propto (-E)^{n-3/2}\)  for bound constituents (and 0 for unbound values). This gives a polytrope with \(\gamma =1+1/n\)  [1]}. For such stellar systems, Ref. [1]} shows that one must have \(n>1/2\)  (\(\gamma <3\) ), but \(n\le 1/2\)  is allowed in general, and I shall consider all values of \(n\)  between 0 and \(\infty \)  (inclusive).
